More Split-Plot Factorial Designs

SPF-pr.q

Linear Model

Y_ijkl = μ + α_j + γ_l + αγ_jl + π_i(jl) + β_k + αβ_jk + βγ_kl + αβγ_jkl + βπ_ki(jl) + ε_ijkl

Let's break the linear model into chunks with an error term at the end of each line.

Y_ijkl = μ + α_j + γ_l + αγ_jl + π_i(jl)       [between subjects]
+ β_k + αβ_jk + βγ_kl + αβγ_jkl + βπ_ki(jl) + ε_ijkl       [within subjects]

Schematic

	b1	b2	b3	b4
a1c1	S1	S1	S1	S1
a1c2	S2	S2	S2	S2
a2c1	S3	S3	S3	S3
a2c2	S4	S4	S4	S4

ANOVA Summary Table (A, B, C fixed)

	Source	Error Term
Between Blocks
1	A	[4]
2	C	[4]
3	A*C	[4]
4	Blks(A*C)
Within Blocks
5	B	[9]
6	A*B	[9]
7	B*C	[9]
8	A*B*C	[9]
9	B*Blks(A*C)
	Total

Stata Command

anova y a c a*c / s|a*c  b a*b b*c a*b*c /, repeated(b)

SPF-p.qr

Linear Model

Y_ijkl = μ + α_j + π_i(j) + β_k + αβ_jk + βπ_ki(j) + γ_l + αγ_jl + γπ_li(j) + βγ_kl + αβγ_jkl + βγπ_kli(j) + ε_ijkl

Again, let's break it into chunks with an error term at the end of each line.

Y_ijkl = μ + α_j + π_i(j)       [between subjects]
+ β_k + αβ_jk + βπ_ki(j)       [within subjects - B]
+ γ_l + αγ_jl + γπ_li(j)       [within subjects - C]
+ βγ_kl + αβγ_jkl + βγπ_kli(j) + ε_ijkl       [within subjects - B*C]

Schematic

	b1 c1	b1 c2	b2 c1	b2 c2
a1	S1	S1	S1	S1
a2	S2	S2	S2	S2

ANOVA Summary Table (A, B, C fixed)

	Source	Error Term
Between Blocks
1	A	[2]
2	Blks(A)
Within Blocks
3	B	[5]
4	A*B	[5]
5	B*Blks(A)
6	C	[8]
7	A*C	[8]
8	C*Blks(A)
9	B*C	[11]
10	A*B*C	[11]
11	B*C*Blks(A)
	Total

Stata Command

anova y a / s|a  b a*b / b*s|a  c a*c / c*s|a  b*c a*b*c /, repeated(b c)

SPF-pr.qt

Linear Model

There are so many terms in this linear model that we will break it up into chunks from the start.

Y_ijkl = μ + α_j + γ_l + αγ_jl + π_i(jl)       [between subjects]
+ β_k + αβ_jk + βγ_kl + αβγ_jkl + βπ_ki(jl)       [within subjects - B]
+ δ_m + αδ_jm + γδ_lm + αγδ_jlm + δπ_mi(jl)       [within subjects - D]
+ βδ_km + αβδ_jkm + βγδ_klm + αβγδ_jklm + βδπ_kmi(jl) + ε_ijklm       [within subjects - B*D]

Schematic

	b1 d1	b1 d2	b2 d1	b2 d2
a1c1	S1	S1	S1	S1
a1c2	S2	S2	S2	S2
a2c1	S3	S3	S3	S3
a2c2	S4	S4	S4	S4

ANOVA Summary Table (A, B, C, D fixed)

	Source	Error Term
Between Blocks
1	A	[4]
2	C	[4]
3	A*C	[4]
4	Blks(A*C)
Within Blocks
5	B	[9]
6	A*B	[9]
7	B*C	[9]
8	A*B*C	[9]
9	B*Blks(A*C)
10	D	[14]
11	A*D	[14]
12	C*D	[14]
13	A*C*D	[14]
14	D*Blks(A*C)
15	B*D	[19]
16	A*B*D	[19]
17	B*C*D	[19]
18	A*B*C*D	[19]
19	B*D*Blks(A*C)
	Total

Stata Command

anova y a c a*c / s|a*c  b a*b c*b a*b*c / b*s|a*c  d a*d c*d a*c*d / d*s|a*c  
  b*d a*b*d c*b*d a*b*c*d /, repeated(b d)

Linear Statistical Models Course